Quods and Quasars
1. The activity
Quod was developed by a computer scientist G. Keith Still in 1979. There are many variations to this game, but the rule of the standard Quod is following:
At the start of the game, each player has 20 attacking pieces - quads (colour) and 6 blocking pieces - quasars (white) depending on the size of the grid.
The objective of the game is to place quads so that they outline a square. The square (called quod) can be of any size and orientation. Player who makes square first is a winner. On each turn player places any number of its quasars and single quad. Quasars are neutral pieces used to occupy places. No cell that has a quasar can be used as a vertex of a square by both players. Quasars can be placed during each player’s term only before they place their quod. The player may decide to place all or none. When both players run out of its quads, player who has more quasars remaining is a winner. Quod is played on 11-by-11 square grid board with the four corners missing.
The following is some variations to this game:
＊Counting quods: Each player plays until all 20 of their quads are placed. Then whoever has the most number of the squares win. If the number of squares is the same, the player with lesser number of quasars used will win.
＊Multiple players: More than two can play with the same rule.
* Team play: Two people form a team and they each place their quods and quasars alternatively. Whichever team that makes the first square or the most square (in case of the counting quods) wins. No direct strategy sharing is allowed among the team members.
Practice board
2.
Problems
2.1 Game related questions
◈ Why did the inventor choose number 11?
◈ Why did he decide to remove four corners?
◈ What is a good strategy for n-by-n grid game?
◈ What is the best first move?
2.2. Mathematical problems
◈ What is the largest square that a player can form?
◈ How many different squares are possible in the grid?
◈ Suppose there are 3 quods that you already have placed. What is the maximum mumber of square you can form by placing the fourth one? What about when you have 4 quods already placed?
2.3. Conjectures
◈ The first player is better off.
◈ For n-by-n grid, there are (n4-n2-48n+84)/12 many possible squares.
◈In 1975, Solomon Golomb raised the question whether the whole plane can be tiled by squares whose sizes are all natural numbers without repetitions, which he called the heterogeneous tiling conjecture.
◈ A "perfect" squared square is a square such that each of the smaller squares has a different size.
The first perfect squared square was found by Roland Sprague in 1939.
If we take such a tiling and enlarge it so that the formerly smallest tile now has the size of the square S we started out from, then we see that we obtain from this a tiling of the plane with integral squares, each having a different size.